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4.7: Use Scale Factor When Problem Solving

Difficulty Level: At Grade Created by: CK-12
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Lifan’s driveway has a length of 24 feet. If the scale is 2 inches:4 feet, what is the scale factor? In a diagram, how many inches would Lifan draw to represent his driveway?

In this concept, you will learn to use scale factors when problem solving.

Scale Factor

The scale can be used to help you with scale dimensions or actual dimensions. This scale is key in problem solving.

If you look at the scale 2:1, you can use this information to determine the scale factor. The scale factor is the relationship between the scale dimension and the measurement comparison between the scale measurement of the model and the actual length. In this case the scale factor is 12.

Let’s look at an example.

What is the scale factor if 3 inches is equal to 12 feet?

First, write the ratio.

312

Next, simplify the fraction.

312=14

The answer is 14.

The scale factor is 1:4.

Now let’s look at a problem where you are applying this information.

If the scale dimension is 4, then you can figure out the actual dimension. Look at this proportion:

1:2=4:x

First, put the proportion in fraction form.

12=4x  

Next, cross multiply to solve for x.

121xx===4x2×48

The answer is 8.

This is the actual dimension.

Let’s look at a real world problem.

The plans for a flower garden show that it is 6 inches wide on the plan. If the scale for the flower garden is 1:12, what is the actual width of the flower garden?

First, write the proportion.

1:12=6:x 

Next, put the proportion in fraction form.

112=6x  

Then, cross multiply to solve for x.

1121xx===6x12×672

The answer is 72.

The actual width of the flower garden is 72 inches.

Examples

Example 1

Earlier, you were given a problem about Lifan and his long driveway. The scale is 2 inches:4 feet and the driveway is 24 feet long.

First, write the proportion. Note that 2:4 is the scale factor.

2:4=x:24

Next, put the proportion in fraction form.

24=x24

Then, cross multiply to solve for x.

244x4x===x242×2448 

Then, divide both sides by 4 to solve for x.

4x4x4x===4848412

The answer is 12.

The scale dimension of Lifan’s driveway is 12 inches.

Example 2

Find the missing actual dimension if the scale factor is 2:3 and the scale measurement is 6.

First, write the proportion.

2:3=6:x

Next, put the proportion in fraction form.

23=6x 

Then, cross multiply to solve for x.

232x2x===6x3×618

Then, divide both sides by 2 to solve for x.

2x2x2x===181829

The answer is 9.

The actual dimension is 9 feet.

Example 3

Find the missing actual dimension if the scale factor is 14:4 and the scale measurement is 8.

First, write the proportion.

14:4=8:x

Next, put the proportion in fraction form.

144116==8x8x

Then, cross multiply to solve for x.

1161xx===8x8×16128

The answer is 128.

The actual dimension is 128 feet.

Example 4

Find the missing actual dimension if the scale factor is 14:4 and the scale measurement is 12.

First, write the proportion.

14:4=x:12

Next, put the proportion in fraction form.

144116==x12x12

Then, cross multiply to solve for x.

11616x16x===x121×1212

Then, divide both sides by 16 to solve for x .

16x16x16xx====121216121634

The answer is 34.

The actual dimension is 34 inches.

Example 5

Find the missing actual dimension if the scale factor is 14:4 and the scale measurement is 16.

First, write the proportion.

14:4=16:x

Next, put the proportion in fraction form.

144116==16x16x

Then, cross multiply to solve for x.

1161xx===16x16×16256

The answer is 256.

The actual dimension is 256 feet.

Review

Figure out each scale factor.

1. 2 inches8 feet

2. 13 inches12 feet

3. 6 inches24 feet

4. 

11 inches33 feet

5. 16 inches32 feet

6. 18 inches36 feet

7. 6 inches48 feet

8. 6 inches12 feet

Solve each problem.

9. A rectangle has a width of 2 inches. A similar rectangle has a width of 9 inches. What scale factor could be used to convert the larger rectangle to the smaller rectangle?

10. A drawing of a man is 4 inches high. The actual man is 64 inches tall. What is the scale factor for the drawing?

11. A map has a scale of 1 inch=4 feet. What is the scale factor of the map?

12. A drawing of a box has dimensions that are 2 inches, 3 inches, and 5 inches. The dimensions of the actual box will be 314 times the dimensions in the drawing. What are the dimensions of the actual box?

13. A room has a length of 10 feet. Hadley is drawing a scale drawing of the room, using the scale factor 150. How long will the room be in Hadley’s drawing?

14. The distance from Anna’s room to the kitchen is 15 meters. Anna is making a diagram of her house using the scale factor of 1150. What will be the distance on the diagram from Anna’s room to the kitchen?

15. On a map of Cameron’s town, his house is 9 inches from his school. If the scale of the map is 1400, what is the actual distance, in feet, from Cameron’s house to his school?

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.7. 

Resources

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    Vocabulary

    Actual Dimension

    The actual dimensions are the real–life measures of the object or building.

    Proportion

    A proportion is an equation that shows two equivalent ratios.

    Scale Dimension

    A scale dimension is the measurement used to represent actual dimensions in a drawing or on a map.

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